On modeling tear breakup dynamics with a nematic lipid layer

Abstract

One of the main roles of the lipid layer (LL) of the tear film (TF) is to help prevent evaporation of the aqueous layer (AL). The LL thickness, composition, and structure all contribute to its barrier function. It is believed that the lipid layer is primarily nonpolar with a layer of polar lipids at the LL/AL interface. There is evidence that the nonpolar region of the LL may have liquid crystalline characteristics. We investigate the structure and function of the LL via a model of the tear film with two layers, using extensional flow of a nematic liquid crystal for the LL and shear-dominated flow of a Newtonian AL. Evaporation is taken into account and is affected by the LL thickness, internal arrangement of its rod-like molecules, and external conditions. We conduct a detailed parameter study with a focus on the evaporative resistance parameter, the Marangoni number, and primary liquid crystal parameters including the Leslie viscosities and director angle. This new model responds similarly to previous Newtonian models in some respects; however, incorporating internal structure via the orientation of the liquid crystal molecules affects both evaporation and flow. As a result, we see new effects on TF dynamics and breakup.

1 Introduction

With each eye blink, a thin liquid tear film is established that helps protect and nourish the ocular surface and helps provide a good smooth surface for clear sight. The tear film (TF) is a multi-layer film, and each layer plays a role in TF dynamics [1]. Despite the importance of the TF, there is still much that is not understood. This paper develops a new model for how the TF may fail, which is often called tear breakup (TBU). The purpose is to better understand the function and structure of the outermost layer of the TF, called the lipid layer (LL). A prime motivation for understanding TF dynamics and TBU is that these phenomena are thought to contribute to the initiation and progression of dry eye disease (DED), which affects millions of people [2, 3]. For comprehensive views of clinical aspects of DED and the tear film see Refs. [4,5,6,7,8]. Reviews related to the tear film include Refs. [1, 9, 10]. In this introduction, we now describe the overall structure of the LL. We then introduce previous work on properties, proceed to mathematical models of the tear film and its dynamics, and finally discuss details of the LL and our goals for this paper.

1.1 Overall structure of the tear film

Figure 1 shows a schematic of part of the TF, which lies on the ocular surface. At the anterior interface of the TF with air is an oily LL that retards evaporation [11, 12], and helps to retain a smooth, well-functioning tear film [13]. The LL also contains polar lipids that are surface active and lower the surface tension between the LL and the aqueous layer (AL), which is posterior to the LL [14, 15]. The AL consists mostly of water [16], but it also contains various ions and proteins [11]. At the corneal surface, which is posterior to the AL, there is a region with transmembrane mucins protruding from the epithelial cells of the cornea or conjunctiva. This forest of glycosolated mucins is called the glycocalyx. It is generally agreed that the presence of the healthy hydrophilic glycocalyx prevents the tear film from dewetting the ocular surface [17,18,19].

Fig. 1

A sketch of the tear film at the corneal surface. Here, LL denotes the lipid layer, AL denotes the aqueous layer, and G denotes the glycocalyx

The overall thickness of the TF is a few microns [20], while the average thickness of the LL is on the order of tens to 100 nanometers when the eye is open [10, 13, 21,22,23]; the thickness of the glycocalyx is a few tenths of a micron [19, 24]. The overall structure is reformed on the order of a second after each blink in a healthy tear film [10, 14, 25, 26].

TBU occurs when a thinned region forms in the TF. Clinically, this is defined as the first dark area that is observed in the fluorescent TF following instillation of fluorescein dye [27]. Various mechanisms are thought to cause different types of TBU, or to combine in various ways: evaporation of water from the AL [5, 7, 28], Marangoni-driven tangential flow (from non-uniform polar lipid surface concentrations) [29], or dewetting-driven tangential flow (from uneven ocular surface conditions) [30,31,32]. Here, we use the phrase tangential flow to mean flow along the eye. The evaporation of water from the AL causes relatively slow thinning [33], while one may expect rapid thinning to be explained by Marangoni-driven tangential flow [29] or dewetting-driven tangential flow. Concentration gradients from surface active polar lipids in the LL induce shear stress at the LL/AL interface and drive outward tangential flow via the Marangoni effect [14, 15]. Dewetting from irregularities in a corneal surface region has been hypothesized to generate outward tangential flow from internal pressure gradients inside the AL due to van der Waals type forces [30, 34, 35]. A related term is full-thickness tear film breakup (full thickness-TBU), which is when the AL has thinned to the point where the LL and glycocalyx touch [28, 36]. It is our belief that the strongest evidence to date regarding mechanisms for TBU are evaporative and Marangoni-driven thinning and their interaction [37, 38]; unambiguous evidence of van der Waals-driven dewetting is not yet in hand in our opinion [28]. We now discuss the studies of these mechanisms in more detail.

The aqueous part of tear fluid is primarily supplied from the lacrimal gland near the temporal canthus (corner) and the excess is drained through the puncta near the nasal canthus [39]. Water lost from the tear film due to evaporation into air is an important process [11, 40, 41]. We believe that this is the primary mechanism by which the osmolarity (essentially, the total concentration of ions on a molar basis) is increased in the tear film [10]. Some water is supplied from the ocular epithelia via osmosis when the osmolarity increases above the isotonic level (the so-called hyperosmolar state) [10, 40, 42,43,44]. According to the Dry Eye WorkShop (DEWS) reports, the osmolarity is important because it plays an important etiological role in the development of dry eye disease [5]. An accepted view in the DEWS reports is that repeated exposure to sufficient hyperosmolarity in tears, as may occur in TBU, can result in inflammation and ocular surface damage in a vicious cycle that may lead to DED. There is still considerable ongoing research to more fully understand the causes and etiology of DED; some mathematical modeling results discussed in this paper help to quantify the relationship between tear breakup and hyperosmolarity.

1.2 Measurements and properties

Both TF instability and hyperosmolarity are important to study because they are proposed as etiological causes of dry eye disease [6, 7]. An osmolarity difference between the corneal epithelium and AL induces osmotic flow from the cornea to the TF [10, 45]. TF osmolarity may be measured in the inferior meniscus clinically [46]; the healthy range is 296–302 mOsM [46,47,48]. Dry eye measurements in the same location can reach 316–360 mOsM [47, 49,50,51] but estimates for the TF over the cornea reach 900 mOsM or higher [10, 45, 52, 53]. High levels of TF osmolarity are associated with pain, inflammation and cellular changes [52, 54, 55]. In support of these potentially high levels of TF osmolarity over the cornea, mathematical models without spatial variation have estimated peak osmolarities up to ten times the isotonic concentration [10, 42]. The modeling work of Peng et al. [45] found that evaporation elevates osmolarity in breakup regions.

TF thinning rates have been measured experimentally or estimated in many studies. A few experimental methods include spectral interferometry [33, 41, 56], an open chamber [57], an evaporimeter [58], and averaging pure water and functioning LL rates over the cornea obtained by heat transfer analysis and thermal imaging [59]. Parameter estimation schemes were developed in Luke et al. [38, 53] for fitting partial differential equation (PDE) models to experimental fluorescent intensity distributions [60]. They found evaporative thinning rates ranging from 36.9 to $-\,4.91$ $\upmu$m/min (negative indicates thickening) and overall TF thinning rates ranging from 23.5 to 1.85 $\mu$m/min. These estimates were quite similar to previous experimental spot-thinning rates that did not specifically seek TBU instances [56].

Tiffany and Dart [61] measured values of the viscosity of human meibum (the material expressed from the meibomian glands inside the lids) using a capillary tube and they estimated values from 9.7 to 19.5 Pa s at 308 K. This is a representative value for the temperature on the surface of the tear film [62]. In a series of papers by Tiffany et al., the human tear film has been shown to have non-Newtonian properties [63,64,65]. Tiffany [65] found that whole tears, that is, tear sampled directly from the eye, are shear thinning. The shear thinning has been fit with empirical models, namely Cross [65] and Ellis models [66]. Tiffany [63] also found weak elastic effects; Tiffany and Nagyová [67] found that removing all lipids from tears caused tear fluid to become Newtonian. The rheology of the LL depends on temperature [68]; in vitro observations showed temperature dependence of meibum on saline for monolayers [69, 70] and for bulk samples [71]. The elastic contributions to viscous response were small above 305 K for these measurements [69]. King-Smith et al. [23] used high-resolution microscopy to observe LL dynamics, and they saw fine structure of the LL during the interblink period that included islands of lipids.

The surface tension of the tear/air interface was measured to be 46.2 mN/m by Miller [72] in vivo using a special contact lens; this is about 2/3 that of an air/pure water interface. Pandit et al. [64] measured the surface tension of whole tears with a capillary tube and obtained 43 mN/m for unstimulated tears and 46 mN/m for stimulated tears; Miller [72] likely measured stimulated tears. A value of 45 mN/m is often used in theoretical studies. The components responsible for lowering the surface tension from that of pure water has been the subject of a number of papers [67, 73,74,75]. A challenge for this measurement is the dependence of surface tension on surface active components like phospholipids and other lipids [76,77,78] and still other species [79]. Detailed equations of state for studying the Marangoni effect from these species are not available for the tear film in vivo to our knowledge, and this is an ongoing area of research.

1.3 Mathematical modeling

A variety of mathematical models have incorporated various important effects of tear film dynamics as has been recently reviewed [10, 42, 80]. The most common assumptions for these models are a Newtonian tear fluid and a flat cornea [14, 81]. One category of models treats the flow over an open eye-shaped domain; these models have used lubrication theory with two space dimensions and time as independent variables [82,83,84,85,86]. These papers were on a stationary domain, and emphasized the effects of supply and drainage, osmolarity in various locations in the tear film, and the overall appearance of fluorescein imaging of the tear film. Some initial studies have examined flow on a blinking domain [80, 87, 88], but they have yet to incorporate the same level of effects as in the static boundary papers. There have also been useful compartment models developed for tear film dynamics [43, 89]. These models use compartments of the tear film as dependent variables in systems of ordinary differential equations (ODE) and thus avoid the computational difficulties of solving PDEs on 2D domains. The models can be solved over many blinks and give insight into the osmolarity and other variables at various operating conditions.

There are quite a few models that have been developed using a single variable in space and time for independent variables. There are two main categories here: (1) a spatial domain that extends across the eye opening vertically, with or without a moving end, and (2) a shorter domain that is a local model for TF dynamics. We first discuss category (1). First, a number of papers considered a domain that extended across the open eye with fixed ends which typically called post-blink drainage models [90,91,92,93]. These models typically imposed large menisci at the domain ends that drained fluid from the relatively flat interior of the TF, and that drove the dynamics. Notable among these papers is that evaporation was included to find that it could drive thinning to TBU [93], and that a moving end, representing the upstroke of the blink, was treated as a coating flow problem [91]. Both of those aspects would find many applications in later papers. Heat transfer and evaporation were considered in Li and Braun [94], and later treated together with thermal imaging in Dursch et al. [59]. A counterpoint to these papers Berger and Corrsin’s treatment of the Marangoni effect in post-blink spreading [14], which provided a basis for using this effect in many later papers.

Using a moving end on a one-dimensional domain to simulate the upper eyelid during a blink has been employed in a number of papers. The first to do this was by Wong et al. [91]; they treated the problem as a coating flow analysis and developed a closed form equation for the thickness of the deposited film as a function of end (lid) speed. Later papers extended this analysis by computing the thickness and other TF variables numerically. These included a combined TF formation and drainage model [95], and then adding surfactant transport on the surface of the TF [96]. Additional papers using the eye opening and interblink together included gravity and reflex tearing [97]; using optimization varying viscosity to simulate and design eyedrops [66]; a detailed look at insoluble surfactant [98]; a (Newtonian) two-layer model of the tear film with insoluble surfactant between them [99]; the effect of (constant) globe curvature on tear film thickness [100]; and a shear-thinning (power law) model for TF thickness on a curved substrate [101]. Some papers also treated the moving end for complete blink cycles. Simple sinusoidal motion was treated in Braun and King-Smith [102], while more realistic motion was treated in Heryudono et al. [103], and insoluble surfactant was added in Zubkov et al. [104]. Heat transfer was later considered with either full [105] or partial [106] blink cycles.

We next discuss category (2) which are local one-dimensional models for TF dynamics. These models are along a short line segment and are designed to be local models of TF flow. The effects of evaporation and the Marangoni effect on TBU have been studied in a number of papers. Evaporation drove all the dynamics in the models in [10, 45, 60], though the treatments were different. In [45], a stationary LL with variable thickness caused increased localized thinning; also included were osmolarity transport in the AL and osmosis across the AL/cornea interface. An important result from that work is that diffusion of osmolarity out of the region of evaporation prevents osmosis from stopping thinning as it would in spatially uniform models [10, 42]. Simpler evaporation models were used in [10, 60], but they included transport of fluorescein so that they could explain what was seen in TF visualization experiments. Zhong et al. [29] developed a PDE model with one spatial variable that incorporated both mechanisms. Including fluorescein dye transport and fluorescence enabled fitting of models to in vivo fluorescence data within TBU regions to estimate parameters that are not possible to directly measure there at the time of writing [38, 53]. Recently, ODE models with no space dependence have been successfully fit to fluorescence data in small TBU spots and streaks [107]. Those models have been coupled to a neural network-based data extraction system to greatly expand the amount of TBU instances that may be studied [108].

Other effects have been included in local models of TBU [29,30,31, 45, 60, 109,110,111,112,113]. Some models incorporated the effect of soluble mucins in the AL necessitating departure from more standard lubrication models [111, 112]. Localized non-wettability of the ocular surface was modeled in [113]. Other models have provided fundamental analyses on evaporation [114, 115] and instability [116, 117].

1.4 LL structure and the model in this paper

The composition and structure of the LL, and their relation to function, are the subject of much ongoing research [76, 118, 119]. The complexity of the composition affects the determination of structure and function and has led to a variety of perspectives in this area [79, 120]. It is generally thought that the LL has an outermost nonpolar LL with a layer of polar lipids and other surface active components at the LL/AL interface [1, 121]. Measuring all but the most basic properties can be a challenge, however. Microscopy has shown some example forms that the LL may take [23]; the shapes observed there suggest non-Newtonian behavior of the LL [70, 71]. For a number of years, it was very difficult to reproduce significant resistance to evaporation using in vitro meibum or LL films [122, 123], even though resistance to evaporation has been thought to be an important property in vivo [11, 33]. However, recent progress on in vitro films with many components has been made [12, 124]. The barrier function does depend on thickness [33, 125], but clearly, there is an effect from internal structure as well [126, 127].

Molecular dynamics simulations, together with Langmuir trough experiments, have also given insight into the structure and function of the LL or model systems for it. Coarse grained models were initially employed [128, 129] and those models suggest that the LL may not be so as well organized as layers of nonpolar lipids atop a polar lipid monolayer. Later, all-atom models have suggested that the specific polar lipids [130, 131] and wax esters [132] may form ordered states to slow evaporation of water from the AL. Molecular dynamics simulations are likely to continue to give important insights into LL structure and function, but they currently can only be solved at small space and time scales. A link to bigger scales has been made via Langmuir trough observations with Brewster angle microscopy, e.g., [130, 131] for equilibrium states. Those insights need to be converted to larger scale dynamic models to address other situations such as TBU. In this paper, we develop a two-layer continuum model that attempts to incorporate some internal structure of the LL that may be expected to produce evaporation resistance [28, 37] that depends on that structure. We use a nematic liquid crystal structure as a model for the LL, which can show birefringence as has been observed for meibum [133].

Adapting two-layer models of tear dynamics [45, 99] to dynamic models of local TBU regions [134, 135] has made use of extensional flow for the LL and shear-dominated flow for the AL. These models have a common basis from the approach of Matar et al. [136]. Of particular interest here is including extensional flow of a nematic liquid crystal [137, 138] for the LL with a Newtonian shear-dominated AL layer. That extensional flow modeling used the Ericksen–Leslie equations and multiple scale perturbation methods to reduce the equations to a small PDE system. A detailed derivation and preliminary results for the combined model were developed in Taranchuk and Braun [135]. With this new model, they found dependence of TF dynamics on the director angle $\theta _B$ with all other parameters fixed, and we continue this work with a detailed parametric study. We find new ways the effects combine to affect TF dynamics and TBU.

The paper is organized as follows. In Sect. 2, we describe the problem formulation. In Sect. 3, we present our results. These include profiles of the film thicknesses, velocities, and pressures that lead to different outcomes depending on the internal orientation of the liquid crystal molecules. Finally, in Sect. 4, we discuss the results and outline our conclusions.

2 Model

Our two-layer model of the tear film lies on top of the corneal surface, which we take to be flat. It is composed of a Newtonian AL containing various solutes and a liquid crystal LL containing non polar lipids. Between the AL and the LL lies a monolayer of surfactant comprising primarily polar lipids. Water in the AL is lost through evaporation, $J_e$, and supplied from the corneal surface through osmosis, $J_o$. Figure 2 shows a sketch of the model. In the AL, we solve for the thickness $h_1$ and the osmolarity concentration c. In the LL, we solve for the following quantities: thickness $h_2$, axial velocity $u_2$, angle of the director field $\theta$, and surface concentration of surfactant $\Gamma$.

Fig. 2

Schematic of the two-layer tear film model

We model the AL using the Navier–Stokes equations, and the LL using the Ericksen–Leslie equations for nematic liquid crystals. We use the following scalings, where primes denote dimensional quantities. The coordinates $(x',y')$ and velocity components $(u_i',v_i')$ correspond to the axial and transverse directions, respectively, $t'$ is time, and $p'_i$ is pressure. The subscript i indicates the fluid layer under consideration; $i=1$ corresponds to the AL, and $i=2$ corresponds to the LL.

$$\begin{aligned} x'&=Lx, \hspace{36pt} h_1'=\epsilon L h_1,\hspace{44pt} h_2'=\epsilon \delta L h_2,\nonumber \\ y'&=\epsilon Ly,\hspace{36pt} u_1'=Uu_1, \hspace{44pt} u_2'= Uu_2, \nonumber \\ h'&=\epsilon L h, \hspace{36pt} v_1'=\epsilon U v_1,\hspace{42pt} v_2'=\epsilon U v_2, \nonumber \\ t'&=\frac{L}{U}t,\hspace{38pt} p_1'= \frac{\mu _1 U}{\epsilon ^2L}p_1,\hspace{33pt} p_2'= \frac{\mu _2 U}{L}p_2, \nonumber \\ \Gamma '&= \Gamma _0\Gamma ,\hspace{38pt} c'=Cc, \hspace{50pt} \nonumber \\ J_o'&= \epsilon U J_o,\hspace{30pt} J_e'=\epsilon \rho _1 UJ_e. \end{aligned}$$

(1)

Here the total thickness of the tear film $h'=h'_1+h'_2\implies h=h_1+\delta h_2$. The scalings used in the model, chosen with the dimension of the human eye in mind, are defined in Table 1. The LL is the thinnest layer in the tear film and has an average thickness of tens to 100 nm during the interblink [13, 22, 139, 140]. The AL is much thicker, averaging about 3 $\upmu$m [141]. The glycocalyx is less than a micron thick [19, 24]; while it does not appear in the model explicitly, we account for its height by defining TBU as an AL thickness of 0.5 $\upmu$m or less. We consider a domain of length of 2 mm.

Table 1 Parameters used in dependent variable scalings

The nondimensional quantities that result from scalings are given in Table 2, and the physical parameters are listed in Table 3. We note that while experiments show evidence of liquid crystal structure in the LL of the TF, the parameter values for the LL have not been measured. For liquid crystals, we return to a Newtonian fluid by setting the Leslie viscosities $\alpha '_i=0,\,i=1,\ldots ,6,\,i\ne 4$, and $\alpha '_4=2\,\mu _2$. The dynamic viscosity of the LL, $\mu _2$, has previously been taken to be 0.1 Pa s [99, 134]. Thus, to match this viscosity in the Newtonian limit, as default values, we choose our Leslie viscosities to be three times the properties of 5CB, a liquid crystal that has been relatively well studied near physiological temperatures [142]. We explore a larger range of values in Sect. 3.1.4.

Table 2 Nondimensional parameters

Table 3 Physical parameters

2.1 Governing equations

After nondimensionalizing, we asymptotically expand all of the dependent variables in powers of the small parameter $\epsilon$. We then collect like powers of $\epsilon$ and solve to find a closed system of leading order equations. In the process, we find that the director angle is constant at leading order, $\theta =\theta _B$. We also determine the leading order AL pressure $p_1$, and the AL depth averaged axial velocity $\bar{u}_1$. This process is described in more detail in the Appendix A and derived thoroughly in Taranchuk and Braun [135]. We find the closed leading order system to be

$$\begin{aligned}&h_{1t}+(\bar{u}_{1}h_{1})_x=J_o-J_e, \end{aligned}$$

(2)

$$\begin{aligned}&h_{2t}+\left( u_{2}h_{2}\right) _x=0, \end{aligned}$$

(3)

$$\begin{aligned}&-\frac{A(\theta _B)}{B(\theta _B)} \delta \Upsilon (u_{2x} h_{2})_x -\frac{u_{2}}{h_{1}}=\frac{h_{1}}{2} \bigg (-{\mathcal {C}}_1 h_{1xxx}- {\mathcal {C}}_2 \Upsilon h_{xxx}\bigg )\nonumber \\&- {\mathcal {C}}_2\delta \Upsilon h_{2} h_{xxx} +{\mathcal {M}} \Gamma _{x}, \end{aligned}$$

(4)

$$\begin{aligned}&\Gamma _{t}+ (\Gamma u_{2})_x=\frac{1}{\text {Pe}_s}\Gamma _{xx}, \end{aligned}$$

(5)

$$\begin{aligned}&(ch_{1})_t =\frac{1}{\text {Pe}_1}(c_{x}h_{1})_x-\left( c\bar{u}_{1}h_{1}\right) _x, \end{aligned}$$

(6)

where

$$\begin{aligned}&h=h_{1}+\delta h_{2}, \end{aligned}$$

(7)

$$\begin{aligned}&J_o={\mathcal {P}}\left( c-1\right) , \end{aligned}$$

(8)

$$\begin{aligned}&J_e= \frac{{\mathcal {E}}}{1+{\mathcal {R}}(\theta _B)h_{2}}, \end{aligned}$$

(9)

$$\begin{aligned}&{\mathcal {R}}(\theta _B)={\mathcal {R}}_0(0.1+0.9\, {\cos \theta _B}), \end{aligned}$$

(10)

$$\begin{aligned}&p_{1}=-\Upsilon {\mathcal {C}}_2h_{xx} - {\mathcal {C}}_1 h_{1xx}, \end{aligned}$$

(11)

$$\begin{aligned}&\bar{u}_{1}=-p_{1x}\frac{h_{1}^2}{12}+\frac{u_{2}}{2}, \end{aligned}$$

(12)

and

$$\begin{aligned} A(\theta _B)=&\;2 \cos (2 \theta _B) (\alpha _1+\alpha _5+\alpha _6+4) (\alpha _2+\alpha _3-\alpha _5+\alpha _6)\nonumber \\&+\cos (4 \theta _B) (\alpha _1 [\alpha _2-\alpha _3]+[\alpha _2+\alpha _3] [\alpha _5-\alpha _6])+\alpha _1 \alpha _2-\alpha _1 \alpha _3-2 \alpha _1 \alpha _5\nonumber \\&-2 \alpha _1 \alpha _6-8 \alpha _1+\alpha _2 \alpha _5+3 \alpha _2 \alpha _6+8 \alpha _2-3 \alpha _3 \alpha _5-\alpha _3 \alpha _6-8 \alpha _3-2 \alpha _5^2\nonumber \\&-4 \alpha _5 \alpha _6-16 \alpha _5-2 \alpha _6^2-16 \alpha _6-32, \end{aligned}$$

(13)

$$\begin{aligned} B(\theta _B)=&-\alpha _1 \cos (4 \theta _B)+\alpha _1-2 \cos (2 \theta _B) (\alpha _2+\alpha _3-\alpha _5+\alpha _6)\nonumber \\&-2 \alpha _2+2 \alpha _3+2 \alpha _5+2 \alpha _6+8. \end{aligned}$$

(14)

Equations (2) and (3) represent conservation of mass (of solvent) in the AL and LL, respectively. Equation (4) is the axial force balance in the LL. Equation (5) conserves mass of surfactant, and Eq. (6) conserves osmolarity in the AL.

The evaporation function in Eq. (9) depends on the evaporative resistance in the LL which is described by Eq. (10). By choosing to implement dependence in this way, smaller director angles provide a stronger barrier to evaporation, and larger director angles provide a weaker barrier to evaporation [37]. This is a representative choice based upon in vitro experiments using molecules with long hydrocarbon chains, similar to those found in the LL of the eye. These experiments show that evaporation resistance is increased when the hydrocarbon tails align in an upright configuration [130, 146, 147]; this corresponds to an angle of $\theta _B=0$ in our model. Further details are given in the Appendix A.3 and in Taranchuk and Braun [135]. We vary the evaporative resistance function ${\mathcal {R}}(\theta _B)$ by changing both ${\mathcal {R}}_0$ and $\theta _B$. Various director angles are considered throughout the paper, and Sects. 3.1.3 and 3.2.2 focus on the effect of ${\mathcal {R}}_0$ for two sets of initial conditions.

2.2 Initial and boundary conditions

2.2.1 Evaporation-driven flow

To illustrate flow driven by evaporation, we choose initial conditions in such a way that valley, or thin spot, in the LL drives all the dynamics of the tear film toward TBU. We choose a Gaussian perturbation of the LL

$$\begin{aligned} h_{2}(x,0)=1-0.9\textrm{e}^{-x^2/(2\sigma ^2)}, \end{aligned}$$

(15)

where $2\sigma ^2=1/9$. We take $h_1=\Gamma =c=1$ initially. We compute $p_1(x,0)$ from Eq. (11), and solve a reformulated version of the axial force balance Eq. (4) using the other initial values to obtain $u_2(x,y,0)$. This initial condition corresponds to the Gaussian disturbance described in Stapf et al. [134]; we refer to it as a valley here.

2.2.2 Marangoni effect

To illustrate surfactant gradient driven flow, we choose initial conditions in which all variables are uniform initially ($h_1=h_2=c=1$), except for a local excess of surfactant in the center of the domain. We compute $p_1(x,0)=0$ and $u_2(x,y,0)$ as described above. The initial condition for $\Gamma$ is

$$\begin{aligned} \Gamma (x,0)=1+s-s\left( \frac{1+0.5\tanh \left( \frac{x-x_0}{x_w}\right) +0.5\tanh \left( \frac{-x-x_0}{x_w}\right) }{n}\right) \end{aligned}$$

(16)

where the height of the region of excess surfactant is $s=1.8$, the inflection point is $x_0=1$, and the width of the region is $x_w=0.36$, and normalization factor is $n=0.9915$. We choose an initial condition of this form because it gives a reasonably large increase in $\Gamma$; if the local excess is too large, the surface tension will become negative. Dimensionally, $x_0$ and $x_w$ are fractions of a mm; this choice is based on observations of the LL in vivo [125]. The normalization n is used for convenience as it gives a background value of unity.

2.2.3 Combining LL and surfactant perturbations

Having considered perturbations in the LL and surfactant surface concentration individually, we now consider two scenarios which combine perturbations in both. Namely, we consider a bump in $h_2$ combined with a local excess of $\Gamma$, and a valley in $h_2$ combined with a local excess of $\Gamma$.

Imaging of the LL has shown instances of regions of thicker lipid [22]. It is reasonable to assume that when the LL is thicker, there is more surfactant. To illustrate this, we consider a thin LL with a central bump and central increased $\Gamma$ in that area. We take $h_1(x,0)=c(x,0)=1$, and we calculate $p_1(x,0)$ and $u_2(x,0)$. The initial condition for surfactant is Eq. (16), and for LL thickness

$$\begin{aligned} h_2(x,0)=1+ \,{ b - b}\left( \frac{1+0.5\tanh \left( \frac{x-x_0}{x_w}\right) +0.5\tanh \left( \frac{-x-x_0}{x_w}\right) }{n}\right) \end{aligned}$$

(17)

where the height of the bump is $b=1$, the inflection point is $x_0=1$, and the width of the bump is $x_w=0.36$, and normalization factor is $n=0.9915$. This corresponds dimensionally to a LL with background thickness of 25 nm, and a bump of lipid of height 50 nm in the center of the domain.

We also consider a pathological scenario in which a valley in the LL is combined with a local excess of $\Gamma$. Keeping all other initial conditions as described above, we modify only Eq. (17) by changing $b=-0.5$, creating a valley in the LL. This corresponds dimensionally to a LL with thickness of 50 nm, and a valley of depth of 25 nm in the center of the domain.

2.2.4 Boundary conditions

At both boundaries, we impose the following no-flux boundary conditions,

$$\begin{aligned} h_{1x}=h_{2x}=p_{1x}=\Gamma _{x}=c_{x}=0, \end{aligned}$$

(18)

and specify $u_2=0$ for the LL velocity.

2.3 Numerical solution

Before solving numerically, we make two change of variables. In order to reduce the order of the system, we define $p_{1}$ as a dependent variable (Eq. (11)) and substitute into Eq. (4). We also use the substitution $m=h_{1}c$ in Eq. (6), where m is the mass. This yields the following closed system which we wish to solve

$$\begin{aligned}&h_{1t}+(\bar{u}_{1}h_{1})_x=J_o-J_e, \end{aligned}$$

(19)

$$\begin{aligned}&h_{2t}+\left( u_{2}h_{2}\right) _x=0, \end{aligned}$$

(20)

$$\begin{aligned} p_{1}&=-\Upsilon {\mathcal {C}}_2h_{xx} - {\mathcal {C}}_1 h_{1xx}, \end{aligned}$$

(21)

$$\begin{aligned}&\quad \,-\frac{A(\theta _B)}{B(\theta _B)} \delta \Upsilon (u_{2x} h_{2})_x -\frac{u_{2}}{h_{1}}=\frac{h_{1}}{2} p_{1x} - {\mathcal {C}}_2\delta \Upsilon h_{2} h_{xxx} +{\mathcal {M}} \Gamma _{x}, \end{aligned}$$

(22)

$$\begin{aligned}&\Gamma _{t}+ (\Gamma u_{2})_x=\frac{1}{\text{ Pe}_s}\Gamma _{xx}, \end{aligned}$$

(23)

$$\begin{aligned}&m_{t} =\frac{1}{\text{ Pe}_1}\left( m_{x}-\frac{mh_{1x}}{h_{1}}\right) _x-\left( m\bar{u}_{1}\right) _x, \end{aligned}$$

(24)

where

$$\begin{aligned}&h=h_{1}+\delta h_{2}, \end{aligned}$$

(25)

$$\begin{aligned}&J_o={\mathcal {P}}\left( \frac{m}{h_{1}}-1\right) , \end{aligned}$$

(26)

$$\begin{aligned}&J_e= \frac{{\mathcal {E}}}{1+{\mathcal {R}}(\theta _B)h_{2}}, \end{aligned}$$

(27)

$$\begin{aligned}&\bar{u}_{1}=-p_{1x}\frac{h_{1}^2}{12}+\frac{u_{2}}{2}. \end{aligned}$$

(28)

We then use the method of lines to solve the system numerically, subject to the initial and boundary conditions of Sect. 2.2. Spatial derivatives are approximated with second-order finite differences on a uniform grid of 512 points. This results in a system of differential algebraic equations that we solve forward in time in Matlab (MathWorks, Natick, MA, USA) using ode15s. We found that the solutions were converged using this number of grid points, and that the mass of water and osmolarity were conserved to the tolerances of the computation.

We integrate the system until either of the following happens: (i) a final time for 60 s is reached, or (ii) TBU occurs. We define TBU as the point at which the AL thickness reaches a minimum value of $0.5\upmu$m and use event detection to stop the simulation if this happens. When breakup occurs, the tear breakup time (TBUT) becomes the final time.

3 Results

3.1 Evaporation-driven thinning

Fig. 3

Solutions for spatially uniform initial conditions except for a Gaussian disturbance in the LL thickness. The director angle is $\theta _B=\pi /2$. This figure illustrates the typical dynamics of the dependent variables in space and time for the case of TBU driven by increased evaporation from the valley in the LL

As an example of evaporation-driven thinning, we consider the initial conditions described in Sect. 2.2.1 with a director angle of $\theta _B=\pi /2$. This choice maximizes the evaporation rate for our chosen evaporation function. Evaporative resistance in the LL is maximized for ${\mathcal {R}}(0)$; ${\mathcal {R}}(\pi /2)$ reduces that resistance by a factor of 10.

Solutions are shown in Fig. 3. The LL changes very little over time, maintaining its original shape with a valley in the center of the domain. The initial velocity in the LL, which ranges between $\pm 50\,\upmu$m, is moderate compared to the dynamics of other scenarios considered later. The thinning in the LL leads to increased evaporation of the AL, which quickly thins, reaching breakup conditions in the center of the domain by 15.9 s. AL pressure begins to drop in the center of the domain; in general, this would help to restore fluid to the area; however, TBU occurs because that capillarity-driven contribution is too weak.

Osmolarity c increases over time, peaking in the center of the domain as a result of increased evaporation in that area. By the final time, osmolarity in the center of the domain is 1651 mOsM, more than five times the initial value; and near the ends, it is 789 mOsM, over two and a half-times the initial value.

The surface surfactant concentration $\Gamma$, shown relative to its initial value, builds up slightly in the center of the domain. We explore Marangoni-driven flow in detail in Sect. 3.2; however, in this example, the increase in surfactant concentration is not large enough to contribute significantly to the flow.

These solutions are representative of evaporation-driven thinning that leads to breakup before the final time of 60 s. With a smaller director angle, evaporation resistance of the LL is increased, and the AL thins more slowly. When $\theta _B\gtrsim 11\pi /25$, TBU is not reached within the simulation time of 60 s. Because evaporation is slower, the valley that develops in the AL is less pronounced, and the resulting peak in osmolarity is smaller; see Taranchuk and Braun [135] for more details.

3.1.1 Comparison with Newtonian case

Fig. 4

Comparison between the liquid crystal case, when $\theta _B=0$ and $\theta _B=\pi /2$, and the Newtonian case. The results for the Newtonian case lie on top of the results for the liquid crystal case when $\theta _B=\pi /2$

In a healthy TF, we expect to have good evaporative resistance from the LL. In our evaporation function Eqs. (9)–(10), as the angle of the director decreases, evaporative resistance increases. Thus, we can compare solutions when $\theta _B=\pi /2$ (shown in detail in the previous section) with those when $\theta _B=0$. When $\theta _B=\pi /2$, we expect to be close to recovering Newtonian viscosity. In our model, we return to the Newtonian case exactly by setting $\alpha '_4=0.2$ Pa s, $\alpha '_i=0,\,i=1,2,3,5,6$, and $\theta _B=\pi /2$. Figure 4 compares the final time solutions for these three scenarios.

We see that as anticipated, the solutions for the Newtonian case lie on top of those for the liquid crystal case when $\theta _B=\pi /2$. This could change if different values for the Leslie viscosities, $\alpha '_i,\,i=1,\ldots ,6$ are used, as discussed more in Sect. 3.1.4. On the other hand, in the liquid crystal case when $\theta _B=0$, there is significantly less thinning in the AL, and TBU is not reached within 60 s. Osmolarity is much lower overall and the peak in osmolarity is small; this is due to a larger AL fluid volume and more time for diffusion of solutes, as compared to the other two cases. In each of the three cases, there is little change in LL thickness and surfactant concentration. Turning to the LL velocity, we see more movement in the two breakup cases, but as breakup conditions are reached quickly, there is little effect on LL thickness.

3.1.2 Varying the Marangoni number

We now consider the effect of the Marangoni number on tear film dynamics as the director angle $\theta _B$ varies from 0 to $\pi /2$. Figure 5 shows the extreme values of AL thickness, osmolarity, and LL thickness for three values of the Marangoni number: ${\mathcal {M}}$ (the default value of 187.7), ${\mathcal {M}}/10$, and $10{\mathcal {M}}$. Solutions are also shown for the initial conditions of Sect. 2.2.2, the local excess of surfactant; these are described in further detail below in Sect. 3.2.1.

We see that when ${\mathcal {M}}\rightarrow {\mathcal {M}}/10$, the Marangoni effect is weakened, and there are no cases of TBU within 60 s. The LL is the most mobile in this case, as healing flow (driven by capillarity) allows the valley in the LL to thicken from its initial minimum of 5 nm, to local minima of 14.7 nm to 33.3 nm, depending on the angle of the director. Maximum osmolarity increases monotonically from 467 to 1323 mOsM.

For the default Marangoni number, TBU occurs before 60 s for $\theta _B \gtrsim 11\pi /25$. The black dot on the peak of $h_{2\textrm{min}}$ marks the separation of these two conditions: to the left of the black dot, no breakup happens before the final simulation time of 60 s; to the right of the black dot, breakup occurs before 60 s and the simulations are stopped at that point. A different definition of TBU would shift the angle at which the peak occurs. Around the critical director angle of $\theta _B = 11\pi /25$ the LL has some mobility, with the lipid valley thickening to a maximum of 17.2 nm; however, for $\theta _B\rightarrow 0$, the LL moves very little from its initial profile.

When ${\mathcal {M}}\rightarrow 10{\mathcal {M}}$, TBU occurs before 60 s for $\theta _B\gtrsim 2\pi /5$. The LL remains tangentially immobile for all angles of the director. Osmolarity increases over the range of the $\theta _B$, but its rise is slower than for the default Marangoni number.

Fig. 5

Extreme values of AL thickness, osmolarity, and LL thickness at the final time as $\theta _B$ is varied from 0 to $\pi /2$, for three values of the Marangoni number ${\mathcal {M}}$. To the right of the black dots, TBUT occurs before 60 s; to the left, the simulations end at 60 s. The first row corresponds to the initial condition with a valley in the LL described in Sect. 2.2.1. The second row corresponds to the initial condition with a local excess of surfactant described in Sect. 2.2.2

3.1.3 Varying the evaporative resistance parameter

Figure 6 shows the effect of varying ${\mathcal {R}}_0$, the evaporative resistance parameter of the LL. The default value ${\mathcal {R}}_0=17.68$ is based on a nonlinear least squares fit to lipid thickness and permeability measurements performed by Stapf et al. [134]; this is shown with the vertical dashed line. With this level of resistance, only a director angle of $\theta _B=\pi /2$ or $\theta _B=4\pi /9$ results in TBUT before 60 s. However, if ${\mathcal {R}}_0$ is decreased by a factor of 10, then TBU occurs in under 15 s for all values of the director angle. As the evaporative resistance parameter is increased, the minimum AL thickness tends toward the initial value of 3.5 $\upmu$m.

Fig. 6

Tear film breakup time or the minimum thickness of the AL after 60 s as a function of the evaporation resistance parameter ${\mathcal {R}}_0$ for the initial condition with a valley in the LL. The black dashed vertical line denotes the default value of ${\mathcal {R}}_0=17.68$

3.1.4 The effect of liquid crystal viscosities on thinning

Fig. 7

Minimum thickness of the AL after 60 s or TBUT as a function of the scale factor applied to the 5CB liquid crystal parameters. Below, $h_{2min}$ as a function of viscosity scale factor. To the right of the black dots, TBUT occurs before 60 s; to the left, the simulations end at 60 s

We have chosen to set the Leslie viscosities to three times the values of 5CB, a relatively well studied liquid crystal. By doing this, we match the dynamic viscosity of the LL, $\mu _2$, to previously used value of 0.1 Pa s [99, 134]. However, there is much uncertainty around the values for the Leslie viscosities as they have not been measured for the LL, and using a scale factor of 3 is only one possible choice. Thus, in Fig. 7, we consider the effect of a wide range of scalings on the minimum values for both the aqueous and LL. Note that because we are scaling the dimensional quantities, in the nondimensional form, we are effectively scaling $\mu _2$. Here, we reduce the Marangoni number by a factor of 10 (${\mathcal {M}}\rightarrow {\mathcal {M}}/10$) in order to isolate the effect of the liquid crystal viscosities on tear film thinning and breakup.

For values of $\mu _2$ that are increased by less than a factor of 1000, there is little change in either the minimum AL thickness or LL thickness at the final time. For values of $\mu _2$ more than a factor of 1000 larger than default, we begin to see breakup within 60 s for large values of $\theta _B$. However, for smaller director angles, the AL is fairly unaffected. On the other hand, for small values of $\mu _2$, we observe a much more mobile LL for all orientations of the director. As $\mu _2$ increases, the LL becomes immobile, and the minimum thickness remains close to the initial value of 5 nm.

Figure 8 shows solutions at the final time when $\mu _2$ is increased by a factor of 2000. At this value, we see the most variation in LL thickness with respect to the director field angle. For clarity, we omit $\theta _B=\pi /4$ and $4\pi /9$. When $\theta _B=0$, the minimum LL thickness is just over 12 nm, while, when $\theta _B=\pi /2$, the effects of capillarity result in healing flow, which increases the minimum thickness to 31.7 nm. For the intermediate value of $\theta _B=5\pi /12$, the LL experiences some healing flow with a minimum of 24.1 nm, but mobility is reduced.

Fig. 8

Solutions for the LL at 60 s when the initial condition has a valley in the LL and $\mu _2\rightarrow 2000\,\mu _2$, and ${\mathcal {M}}\rightarrow {\mathcal {M}}/10$

Fig. 9

The coefficient A/B as $\theta _B$ varies for scalings of 3, 10, and 16 on the Leslie viscosities $\alpha {'}_i,\,i=1,\ldots ,6,\,i\ne 4$ for the liquid crystal 5CB

It is natural to explore an alternative scenario: keeping a scaling of three on $\alpha '_4$, which maintains $\mu _2=0.1$ Pa s, while scaling the other Leslie viscosities, $\alpha '_i,\,i=1,\ldots ,6,\,i\ne 4$. By doing this, we are changing the coefficient A/B on the extensional term in Eq. (4). Note that as $\alpha '_i\rightarrow 0,\,i=1,\ldots ,6,\,i\ne 4$, we return to the Newtonian limit, where $A/B=4$. However, we find that if we increase the scaling too much (more than a factor of 16), the coefficient A/B becomes positive for some values of $\theta _B$, making the problem ill-posed. In addition, the effect of the scalings in this feasible range is negligible for all variables. If we look at the coefficient A/B for scalings, shown in Fig. 9, we see that the change on the coefficient is very little, even with the maximum scale factor of 16.

3.2 Marangoni-driven thinning

We now turn to the initial conditions of Sect. 2.2.2, in which concentration gradients drive the tear film dynamics. The Marangoni effect results in rapid tangential flow; thus we consider solutions on a short timescale, shown in Fig. 10, and a longer timescale, shown in Fig. 11. Figure 10 shows the evolution of the variables over the first 0.06 s, when $\theta _B=0$. These solutions are representative of the dynamics of tear film thinning due to surfactant concentration gradients regardless of the orientation of the liquid crystal molecules; the director angle $\theta _B$ has little effect on the short timescale. The large surfactant gradients immediately cause rapid thinning in both the lipid and the aqueous layers. As the local excess of surfactant flattens, the Marangoni effect concludes. The LL thins dramatically, with the valley thickness roughly half of the initial value of 50 nm. Fluid is pulled away from the excess surfactant and builds up on the edges of the domain. The pattern of thinning is very similar in the AL, although to a lesser degree. During this initial redistribution of fluid, the LL fluid velocity is very large in magnitude (around $3\times 10^5\,\upmu$m/min), but quickly slows to the range of $\pm 5\,\upmu$m/min throughout the domain (not shown). Similarly, AL pressure (not shown) drops in magnitude from a range of $\pm 3$ Pa during the first 1 s, to $\pm 1$ Pa, and thinning continues over a longer timescale.

Fig. 10

Solutions on a short time scale for spatially uniform initial conditions except for a local excess of surfactant. The director angle is $\theta _B=0$. This figure illustrates the typical dynamics of the dependent variables in space and time. The surface gradients of surfactant drive all the dynamics

Fig. 11

Solutions at the final time for spatially uniform initial conditions except for a local excess of surfactant. When $\theta _B=\pi /2$, breakup occurs at 16.6 s; for all other values of $\theta _B$, the final time is 60 s. The LL profiles (upper right) are nearly identical for all $\theta _B$

Figure 11 shows solutions at the final time for fives values of $\theta _B$ ranging from 0 to $\pi /2$. Breakup conditions are reached when $\theta _B=\pi /2$ by 16.6 s; for all other values of $\theta _B$, the final time of 60 s is reached. The surfactant concentration remains effectively uniform around 1.58 times the original background concentration, and over the longer timescale, it is the evaporative resistance in the LL that influences thinning dynamics. When $\theta _B=0$ or $\pi /4$, thinning becomes more localized in the center of the domain, while larger values of $\theta _B$ maintain a wider valley of thinning through time. For smaller values of $\theta _B$, we observe that negative pressure in the center of the AL domain pulls fluid in and helps to reduce thinning. This results in an overall thicker AL at the final time. Arrows in the velocity and pressure plots of Fig. 11 point to the difference in behavior when $\theta _B=0$ or $\pi /2$ as compared to smaller director angles.

On the other hand, the LL profiles are indistinguishable from each other at the final time and look very similar to the solution at 0.06 s shown in Fig. 10. Virtually all of the movement in the LL happen within the first second; this immobilization of the fluid surface due to the Marangoni effect has been studied in other contexts such as vertical drainage in Braun et al. [148] and remobilization of droplet surfaces in Stebe and Maldarelli [149].

Osmolarity in the AL increases for all values of $\theta _B$ over the original value of 300 mOsM; however, it is most pronounced when $\theta _B=\pi /2$. While the solutions in the AL are not dissimilar for $\theta _B=4\pi /9$ and $\theta _B=\pi /2$, the osmolarity is very different. When $\theta _B=\pi /2$, peak osmolarity reaches nearly 1400 mOsM by 16.6 s, more than one and a half-times the peak osmolarity at 60 s when $\theta _B=4\pi /9$. When $\theta _B=\pi /2$, thinning and TBU happen so rapidly that the solutes do not have enough time to diffuse through the AL.

3.2.1 Varying the Marangoni number

The second row of Fig. 5 shows extreme values for AL thickness, osmolarity, and LL thickness as the director angle varies from 0 to $\pi /2$ for three values of the Marangoni number: the default ${\mathcal {M}}$ (187.7), ${\mathcal {M}}/10$, and $10{\mathcal {M}}$. In general, compared to the initial condition with a valley in the LL which are discussed in Sect. 3.1.2, varying the Marangoni number has a much smaller effect on the variables. This is because with localized excess surfactant in the initial condition, the Marangoni effect is already driving dynamics, and our default Marangoni number is strong enough that increasing or decreasing by a factor of 10 does not have a dramatic effect.

For all three values of the Marangoni number, TBU occurs for $\theta _B\gtrsim 12\pi /25$ when the initial condition has a local excess of surfactant. For smaller director field angles, there is no breakup within 60 s, but the AL thins much more than in the case with the lipid valley, with minimum AL thickness ranging from 1.62 to 1.84 $\upmu$m when $\theta _B=0$. After the initial rapid redistribution of fluid due to the Marangoni effect, the valley formed in the LL has a minimum value of 28 nm when $\theta _B=\pi /2$ for all three cases. For ${\mathcal {M}}$ and $10{\mathcal {M}}$, the minimum remains the same for all values of the director. For ${\mathcal {M}}/10$, the minimum increases slightly as the director angle increases, with a minimum value of 31.6 nm when $\theta _B=0$. Osmolarity between the three cases is very similar, with a maximum difference of 28 mOsM; in fact, the osmolarity curve for $10{\mathcal {M}}$ is indistinguishable from that of ${\mathcal {M}}$ in Fig. 5.

3.2.2 Varying the evaporative resistance parameter

We now explore the effect of the evaporative resistance parameter ${\mathcal {R}}_0$. Figure 12 shows the TBU conditions as ${\mathcal {R}}_0$ varies for various orientations of the director. A small evaporative resistance parameter results in rapid thinning of the AL with breakup just under 5 s for all values of $\theta _B$. As ${\mathcal {R}}_0$ increases, breakup time increases, until eventually the final time of 60 s is reached before breakup occurs. The value of ${\mathcal {R}}_0$ at which this occurs varies from 2.5 for $\theta _B=0$ to 25.1 for $\theta _B=\pi /2$. The default value, denoted by the dashed vertical line, falls into this range where thinning may result in breakup before 60 s, depending on the director angle. See Fig. 11 for solutions at the final time. As ${\mathcal {R}}_0$ becomes very large, evaporation is diminished, and the minimum value tends toward 3.1 $\upmu$m for all values of $\theta _B$.

Fig. 12

Tear film breakup time or the minimum thickness of the AL after 60 s as a function of the evaporation resistance parameter ${\mathcal {R}}_0$ for the initial condition that has a local excess of surfactant. The black dashed vertical line denotes the default value of ${\mathcal {R}}_0=17.68$

Figure 13 shows the relationship between the evaporative resistance parameter and the angle of the director for the two sets of initial conditions. The value of ${\mathcal {R}}_0$ which results in breakup conditions at 60 s is shown in red for $\theta _B=0,\,\pi /4,\,5\pi /12,\,4\pi /9,$ and $\pi /2$. The circles represent the solution from the evaporation-driven flow caused by a valley in the LL; the crosses represent the solutions from flow driven by the Marangoni effect caused by localized excess surfactant. The black lines are exponential functions fit to the respective data delineating whether or not breakup conditions are reached by the final time of 60 s. If the evaporative resistance parameter is decreased by a factor of 10 from the default value of 17.68, then TBU will occur for all angles of the director for both sets of initial conditions, although TBUT will vary.

If we compare the two curves, we see that the ${\mathcal {R}}_0$ which results in TBU at 60 s for the case of evaporation-driven thinning is roughly double that of the case of Marangoni-driven flow. In the evaporative case, when $\theta _B=\pi /2$, the evaporative resistance parameter ${\mathcal {R}}_0=50$, whereas for the Marangoni-driven thinning, ${\mathcal {R}}_0=25$. When $\theta _B=0$, ${\mathcal {R}}_0$ is 5 and 2.5, respectively. This trend roughly holds for all values of the director angle. In general, this is expected behavior; the LL acts as a barrier to evaporation in part due to its thickness. In the first set of initial conditions, a deficiency in the LL leads to increased evaporation. Thus, it cannot tolerate an evaporation resistance parameter value of any less than 50 without TBU occurring before 60 s when $\theta _B=\pi /2$. On the other hand, in the Marangoni-driven flow, the LL begins with a uniform thickness of 50 nm which helps to prevent TBU even though other aspects of the barrier may be compromised.

Fig. 13

A comparison of the ${\mathcal {R}}_0$ value which results in TBU at 60 s for two sets of initial conditions: one describing evaporation-driven flow (red circles), and one describing flow driven by the Marangoni effect (red crosses), as a function of the director angle of the liquid crystal molecules in the LL. The black lines are a fit to the respective data. To the right of each black line, TBU occurs before 60 s; to the left of each black line, TBU may occur after 60 s

3.2.3 Reducing initial lipid layer thickness

The thickness of the LL ranges between 20 and 120 nm, with more recent measurements favoring a thinner LL [22, 139, 140]. We halve the initial LL thickness from 50 nm (a “normal” tear film thickness) to 25 nm (a “thin” thickness). Figure 14 shows the solutions for the AL and osmolarity at the final time for a few values of $\theta _B$. The solutions can be compared with those of Fig. 11 which show solutions for a 50 nm thick initial LL. Qualitatively, solutions for small values of $\theta _B$ are similar, although the AL is thinner, especially at its minimum. However, for $\theta _B=5\pi /12$ or $4\pi /9$, there are two minima, located symmetrically about the center. For $\theta _B=\pi /2$, the region of thinning is steeper and more narrow than when the LL is thicker. In all cases, the solutions for osmolarity show higher concentrations with a thinner LL.

Table 4 shows that a 50% decrease in LL thickness results in a 40.1% decrease in the minimum AL thickness when $\theta _B=\pi /4$; for other values of the director field angle, the percent decrease is even less. The factor increase in peak osmolarity over the ideal osmolarity is also shown. These numbers are highest when TBU occurs; however, with a thinner LL, peak osmolarity is well over twice the ideal value for all angles of the director.

Table 4 Comparison of minimum AL thickness and maximum osmolarity when the initial LL thickness is reduced from 50 to 25 nm

Fig. 14

Solutions at the final time for spatially uniform initial conditions except for a local excess of surfactant. The initial thickness of the LL is 25 nm (half the default value). When $\theta _B=\pi /2$, breakup occurs at 8.2 s; when $\theta _B=4\pi /9$, breakup occurs at 54.8 s. For all other values of $\theta _B$, the final time is 60 s

Figure 15 shows the effect of varying the director angle by plotting parametric changes to the minimum value of the AL thickness and the maximum value of osmolarity at the final time as $\theta _B$ varies from 0 to $\pi /2$ for both a normal and thin LL. When the LL thickness is 50 nm, TBU occurs before 60 s only for values of $\theta _B>12\pi /25$; when the LL thickness is 25 nm, TBU occurs for $\theta _B>3\pi /7$. The black dots on the plot mark this change. Minimum LL thickness as $\theta _B$ varies is not shown, as changes are minimal. When the initial LL thickness is 50 nm, minimum values hovers near 28 nm; when the initial thickness is 25 nm, the minimum values remain around 14 nm for all values of the director angle.

Fig. 15

Extreme values of AL thickness and osmolarity at the end of each simulation as $\theta _B$ is varied from 0 to $\pi /2$. To the right of the black dot, TBUT occurs before 60 s; to the left of this dot, the simulations end at 60 s

3.3 Combining lipid layer and surfactant perturbations

We now compare the two scenarios which combine perturbations in both the LL thickness and the surfactant concentration. The first combination is a bump in $h_2$ with a local excess of $\Gamma$, and the second is a valley in $h_2$ with a local excess of $\Gamma$. Figure 16 compares the solutions for AL and LL thickness, as well as osmolarity for these two cases with $\theta _B=0$. While neither case reaches breakup for this angle of the director, thinning in the AL is more pronounced when there is a valley in the LL, with a minimum value of 0.94 $\upmu$m, as compared to 1.66 $\upmu$m when the LL contains a bump. The LL behavior is similar in both cases; it undergoes rapid redistribution of fluid initially and then remains tangentially immobile for the remainder of the time. When the LL contains a bump, rapid redistribution of fluid throughout the domain results in an area of thinning where the bump was previously located, with the minimum thickness of 28.46 nm. When the LL contains a valley, the Marangoni effect magnifies this profile resulting in a solution similar to those of Fig. 11, but with a deeper valley: the depth is now 14.15 nm versus 28.1 nm (when $\theta _B=0$).

Osmolarity in both cases reflects the respective changes in the AL. When the LL contains a valley, the AL thinning is more pronounced, and thus osmolarity reaches a peak of 650 mOsM, with little increase in concentration at the edges of the domain. When the LL contains a bump, the AL thins to a lesser degree, and the osmolarity steadily rises throughout the domain, with a smaller peak in the center. Peak osmolarity at the final time reaches 485 mOsM. In all scenarios that contain a local excess of surfactant initially, the Marangoni effect results in a rapid flattening of the surfactant concentration to a more or less uniform value of 1.58.

We can consider the influence of the director field angle, by observing the final time solutions for various angles as shown in Fig. 17. When there is a bump in the LL, we see more variation resulting from changes in the director angle. When $\theta _B=\pi /2$, TBU occurs at 46.8 s, but no other angles of the director result in breakup. When there is a valley in the LL, the director angle has less of an impact because rapid thinning in the AL leads to TBU for $\theta _B=5\pi /12,\,4\pi /9$, and $\pi /12$. Breakup times are detailed further in Table 5 which is discussed more below. Osmolarity at the final time is strongly influenced by the director angle. When there is a bump in the LL, peak osmolarity is almost three times higher for $\theta _B=\pi /2$ (1409 mOsM) than for $\theta _B=0$ (485 mOsM). The difference is just over double when there is a valley in the LL (1442 versus 650 mOsM).

Thus far we have considered a valley in the LL with depth of 25 nm in a 50 nm LL. We now vary this profile keeping a valley of half-thickness. Table 5 compares the TBUT for three different LL background thicknesses: 30, 40, and 50 nm. We see that the dynamics are not sensitive to LL thickness when $\theta _B=\pi /2$. This is because evaporative resistance is at its lowest for this angle of the director; for all of the scenarios, we have considered $\theta _B=\pi /2$ results in TBU before 60 s. LL thickness has a stronger effect for intermediate angles of the director. For example, when $\theta _B=4\pi /9$, decreasing the LL thickness by 20%, from 50 to 40 nm, results in TBU in half the time. When $\theta _B$ is small enough, TBU does not occur; for all three LL thicknesses, TBU does not occur for $\theta _B=0$ or $\pi /4$,

Fig. 16

Left column: Solutions when $\theta _B=0$ for a bump in $h_2$ combined with a local excess of $\Gamma$. Right column: Solutions when $\theta _B=0$ for a valley in $h_2$ combined with a local excess of $\Gamma$. In both cases, $\Gamma (x,0)$ is given by Eq. (16). The different $h_2(x,0)$ appears in the middle row

Fig. 17

Left column: Solutions at the final time for a bump in $h_2$ combined with a local excess of $\Gamma$. Right column: Solutions at the final time for a valley in $h_2$ combined with a local excess of $\Gamma$

Table 5 TBU time comparison in seconds for various initial LL thicknesses when a valley in the LL is combined with a local excess of surfactant

4 Discussion and conclusion

In this paper, we explore a new model for the tear film of the eye that was derived in Taranchuk and Braun [135]. This two-layer model combines a Newtonian AL with a nematic liquid crystal LL and links molecular orientation of the LL with evaporative resistance that depends on the director angle. The LL acts as a barrier to evaporation of the AL, and thickness alone does not explain its role in preventing evaporation [126, 127]. By representing the LL as nematic liquid crystal, we are able to consider possible connections between the structure and composition of the LL and its affect on tear film thinning and breakup.

We examined two key drivers of TBU: evaporation and the Marangoni effect. We consider several scenarios which may lead to TBU. The first is a narrow valley in the LL, which leads to thinning of the AL due to increased evaporation. The second is a local excess of surfactant on the AL/LL interface, for which we see thinning on two timescales. Initially, surfactant gradients lead to rapid outward flow in both the AL and LL. The surfactant concentration levels out within the first second, and then evaporation takes over, continuing tear film thinning over a longer timescale of 60 s. Finally, we considered combinations of perturbations in both the LL and surface surfactant; one in which there is a bump in the LL combined with a local excess of surfactant and one in which there is a valley in the LL combined with a local excess of surfactant. This final case depicts a worst case scenario, where both excess surfactant and a deficiency in the LL drive thinning, and breakup conditions are reached for $\theta _B\gtrsim 2\pi /5$.

The evaporative resistance parameter ${\mathcal {R}}_0$ directly influences how quickly the AL thins. However, as shown in Fig. 13, the minimum value of ${\mathcal {R}}_0$ that the system can tolerate while withstanding breakup also depends on the angle of the director field, $\theta _B$, and the initial condition for the LL. The evaporation function, given in Eqs. (9) and (10) and derived in A.3, shows one way these quantities may be related. While we have chosen to introduce director angle dependence in the evaporation function [28, 37, 146], we also see that the director angle has an effect on flow dynamics. Large angles of $\theta _B$ cause the LL to become more mobile, which results from liquid crystal properties in the LL. When $\theta _B=\pi /2$, the results are very close to those in the Newtonian case.

In addition, we find that the default Marangoni number has a strong effect on dynamics, regardless of the initial conditions or other parameter values used. Thus, in order to observe the effect of the liquid crystal parameters, especially the Leslie viscosities, we reduce the Marangoni number by a factor of ten. In the absence of any measurements in the eye, we use the properties of 5CB as a starting point for the Leslie viscosities of the liquid crystal lipid layer. One unexpected finding is that varying the Leslie viscosities from the 5CB values has little effect on the dynamics. Not only that, but also we are constrained in how much we can vary them, at least when using constant scalings of 5CB. On the other hand, the other main liquid crystal parameter $\theta _B$, which describes the orientation of the molecules, affects both evaporation (by design) and flow. This is especially true for small angles.

Rather than varying all of the Leslie viscosities, we focused on varying $\alpha '_4$, which controls the dynamic viscosity of the LL (because $\mu _2=\alpha '_4/2$). These results are very similar to the Newtonian case, explored in Stapf et al. [134], in that it takes a very large increase in $\mu _2$ to effect the dynamics of the AL. The director angle does have an effect on the mobility of the LL, until $\mu _2$ becomes very large (on the order of $10^5$). After this point, the LL becomes virtually tangentially immobile for all angles of the director angle.

Finally, we consider two initial conditions which combine perturbations in both the LL and the surface surfactant. The first is an undesirable case where a valley of lipid is combined with a local excess of surfactant. As expected, we see more breakup cases; for all angles of $\theta _B\gtrsim 2\pi /5$, TBU occurs within 60 s as a result of both the Marangoni effect and increased evaporation due to the deficiency in the LL.

Alternatively, we consider an initial condition which combines a bump in the LL with a local excess of surfactant. This scenario corresponds to Sect. 5.6 in Bruna and Breward [99], where in a limit $h_2$ and $\Gamma$ can both be governed by the same equation. While we do not take that limit here, we model an initial condition in which the LL follows the surfactant. This area of increased thickness in the LL seems to be protective by creating a buffer against the Marangoni effect, and thus helping to prevent excessive thinning in the AL. Imaging of the LL has shown areas of varying thickness on occasion [22]. This type of initial condition could resemble this scenario, but more work is needed to link them.

Choudhury et al. [113] modeled TBU driven by dewetting due to loss of mucins in the glycocalyx using a single-layer model. They consider 119 s to be a normal time to breakup in a healthy eye, and their simulation resulted in TBU as short as 3.3 s when there is mucin loss over a third of the corneal surface. Dey et al. [111, 112] studied the same mechanism for TBU, but used a continuous-viscosity model which combined the mucus and aqueous layers. Their model predicted breakup in 2.4 to 135 s for scenarios representing pathological to healthy conditions, respectively. In vivo measurements for subjects with dry eye or other eye conditions have shown TBU times well under 10 s. In healthy subjects, TBU times may be longer, but would generally occur in less than two minutes [32, 36, 125].

While our model focuses on different mechanisms for thinning and breakup, TBU times from our simulations fall within a similar range as previous models and experiments. We consider that a healthy eye has high evaporative resistance, corresponding to a small director field angle. For the scenarios we considered, a director angle of $\theta _B=0$ or $\pi /4$ did not result in breakup within 60 s. We did not compute time to TBU; however, at the end of 60 s, minimum AL thicknesses were well above the threshold value of 0.5 $\upmu$m for these smaller director angles. For large angles of the director in which evaporative resistance was low and there was a deficiency in the LL or an excess of surfactant, our model predicted breakup in as little as 6 to 10 s. These predicted breakup times are not as rapid as the shortest TBU times observed in eyes [32, 125].

Thinning and breakup due to a glob of surfactant were studied in Zhong et al. [29, 110]. The glob consisted of an interval of fixed concentration as well a part that was transported outside this interval. This area of fixed concentration, which acted as a source, was varied in size and found to cause TBU either under or at the edge of the glob depending on size. TBU occurred in $< 1$ s to 15 s, and it was noted that evaporation aided TBU. Our results have central breakup and evaporation aligns with the Marangoni effect to promote TBU. Although our TBU time is on the longer end of the range they predicted, our treatment of surfactant is simpler. We conserve surfactant, and the local excess of surfactant does not act as a source; nor did we experiment with larger or smaller concentrations.

Thinning due to evaporation has been modeled previously both with a fixed evaporation distribution [60], and with a static LL [45]. Alternatively, Luke et al. [53] developed an evaporation-driven model that was fit to experimental data. We have taken a mechanistic approach, and incorporated LL thickness, permeability, and structure in our evaporation function.

We have studied a variety of initial conditions and explored the effects of several key parameters on tear film dynamics when modeling the LL as nematic liquid crystal. Now that we have a sense of the key features and limitations of this model, and future work could include a moving end to imitate the eyelid during a blink. Tear breakup is often related to the LL pattern [10, 125]; more fidelity would require a moving domain [96]. In addition, it would be valuable to determine more rheological properties for this kind of model [69,70,71].

Dimensional equations

We give the dimensional equations (after [135]) and a very brief outline of the derivation.

1.1 Governing equations

Inside the AL, $0<y'<h'_1$, we have conservation of mass and momentum using the Navier–Stokes equations. The advection–diffusion equation governs osmolarity transport within the AL. The equations in dimensional form with a prime denoting a dimensional quantity, respectively, are:

$$\begin{aligned}&\nabla ' \cdot \mathbf{u'}_1=0, \end{aligned}$$

(A1)

$$\begin{aligned}&\rho _1 (\mathbf{u'}_{1t{'}}+\mathbf{u'}_1\cdot \nabla ' \mathbf{u'}_1)= -\nabla ' p'_1 + \mu _1 \Delta ' \mathbf{u'}_1, \end{aligned}$$

(A2)

$$\begin{aligned}&c'_{t'} + \mathbf{u'}_1\cdot \nabla ' c' - D_1 \Delta ' c'=0, \end{aligned}$$

(A3)

where the divergence of $\mathbf{u'_1}$ is $\nabla '\cdot \mathbf{u'_1}=u'_{1x'}+v'_{1y'}$, the gradient is $\nabla '\mathbf{u'_1}=\begin{bmatrix} u'_{1x'} &{}v'_{1x'}\\ u'_{1y'}&{} v'_{1y'}\end{bmatrix}$, and the Laplacian of the first component of $\mathbf{u'_1}$ is $\Delta ' u'_1=u'_{1x'x'}+u'_{1y'y'}$.

Inside the LL, $h'_1<y'<h'$, we have conservation of energy, mass, and momentum using the Ericksen–Leslie equations for liquid crystals.

$$\begin{aligned} \frac{\partial }{\partial x'}\left( \frac{\partial W'}{\partial \theta _{x'}}\right) +\frac{\partial }{\partial y'}\left( \frac{\partial W'}{\partial \theta _{y'}}\right) -\frac{\partial W'}{\partial \theta }+{\tilde{g}}'_x\frac{\partial n_x}{\partial \theta }+{\tilde{g}}'_y\frac{\partial n_y}{\partial \theta }&=0, \end{aligned}$$

(A4)

$$\begin{aligned} -\frac{\partial \pi '}{\partial x'}+{\tilde{g}}'_x\frac{\partial n_x}{\partial x'}+{\tilde{g}}'_y\frac{\partial n_y}{\partial x'}+\frac{\partial {\tilde{t}}'_{xy}}{\partial y'}+\frac{\partial {\tilde{t}}'_{xx}}{\partial x'}&=0, \end{aligned}$$

(A5)

$$\begin{aligned} -\frac{\partial \pi '}{\partial y'}+{\tilde{g}}'_x\frac{\partial n_x}{\partial y'}+{\tilde{g}}'_y\frac{\partial n_y}{\partial y'}+\frac{\partial {\tilde{t}}'_{yx}}{\partial x'}+\frac{\partial {\tilde{t}}'_{yy}}{\partial y'}&=0, \end{aligned}$$

(A6)

$$\begin{aligned} \frac{\partial u_2'}{\partial x'}+\frac{\partial v_2'}{\partial y'}&=0, \end{aligned}$$

(A7)

where, working in two dimensions,

$$\begin{aligned} {\tilde{g}}'_i=&-\gamma _r N'_i-\gamma _t \textrm{e}'_\mathrm{{ik}}n_k, \hspace{25pt} \textrm{e}'_{{ij}}=\frac{1}{2}\left( \frac{\partial u'_i}{\partial x'_j}+\frac{\partial u'_j}{\partial x'_i}\right) , \end{aligned}$$

(A8)

$$\begin{aligned} N'_i=&\;{\dot{n}}'_i-\omega '_{{ik}}n_k, \hspace{55pt} \omega '_{ij}=\frac{1}{2}\left( \frac{\partial u'_i}{\partial x'_j}-\frac{\partial u'_j}{\partial x'_i}\right) ,\hspace{25pt}\pi '=\;p_2'+W', \end{aligned}$$

(A9)

$$\begin{aligned} W'=&\;\frac{1}{2}\bigg [K_1(\nabla ' \cdot \textbf{n})^2+K_3((\textbf{n}\cdot \nabla ') \textbf{n})\cdot ((\textbf{n}\cdot \nabla ') \textbf{n})\bigg ], \end{aligned}$$

(A10)

$$\begin{aligned} {\tilde{t}}'_{ij}=&\;\alpha _1'n_kn_p\textrm{e}'_{kp}n_in_j+\alpha '_2N'_in_j+\alpha '_3 N'_jn_i+\alpha '_4\textrm{e}'_{ij}+\alpha '_5\textrm{e}'_{{ik}}n_kn_j+\alpha '_6\textrm{e}'_{{jk}}n_kn_i . \end{aligned}$$

(A11)

We use summation notation here as it is standard when working with the Ericksen–Leslie equations, with $i,j,k=1,2$, and ${\dot{n}}_i$ denotes the convective derivative of the ith component of $\textbf{n}$. We further assume that the elastic constants are equal, that is $K=K_1=K_3$. These liquid crystal quantities are defined further in Table 6.

Table 6 Liquid crystal variables and parameters [142]

1.2 Boundary conditions

We now list the boundary conditions, proceeding from the cornea out toward the air. At $y'=0$, we have velocity continuity and a no-flux boundary condition for osmolarity transport. These are, respectively:

$$\begin{aligned}&\mathbf{u'_1} = \mathbf{J_0}, \end{aligned}$$

(A12)

$$\begin{aligned}&c'\,(\mathbf{u'}_1 - \mathbf{s'}_{0t'})-D_1 \nabla ' c'=0. \end{aligned}$$

(A13)

Note that as we assume the cornea is flat, $\mathbf{s'}_{0t'}=\textbf{0}$.

At $y'=h'_1$, we have velocity continuity, aqueous and lipid mass conservation, and the stress balance. In addition, we have a no-flux boundary condition for osmolarity transport, and the anchoring boundary condition for the LL. The equations, respectively, are:

$$\begin{aligned}&(\mathbf{u'}_1 - \mathbf{u'}_2)\cdot {\hat{t}}'_1 = 0, \end{aligned}$$

(A14)

$$\begin{aligned}&\rho _1 (\mathbf{u'}_1 - \mathbf{s'}_{1t'}) \cdot {\hat{n}}'_1 = J'_e, \end{aligned}$$

(A15)

$$\begin{aligned}&\rho _2 (\mathbf{s'}_{1t'}-\mathbf{u'}_2) \cdot {\hat{n}}'_1 = 0, \end{aligned}$$

(A16)

$$\begin{aligned}&(\mathbf{\tau '}_1 - \mathbf{\tau '}_2)\cdot {\hat{n}}'_1 = -\gamma '_{s1}{\hat{n}}'_1 \nabla ' \cdot {\hat{n}}'_1 + \nabla '_{s_1} \gamma '_{s1}, \end{aligned}$$

(A17)

$$\begin{aligned}&c'\,(\mathbf{u'}_1 - \mathbf{s'}_{1t'})-D_1 \nabla ' c'=0, \end{aligned}$$

(A18)

$$\begin{aligned}&\theta =\theta _B. \end{aligned}$$

(A19)

To resolve the stress balance into tangential and normal stress components, we take the dot product of the stress balance and ${\hat{t}}'_1$ or ${\hat{n}}'_1$, respectively. Here $\nabla '_{s_1}=(\textbf{I}-{\hat{n}}'_1{\hat{n}}'_1)\cdot \nabla '$ is the surface gradient, where $\textbf{I}$ is the identity matrix [150].

At $y'=h'$, we have lipid mass conservation, stress balance, and anchoring, respectively:

$$\begin{aligned}&\rho _2 (\mathbf{u'}_2 - \mathbf{s'}_{2t'})\cdot {\hat{n}}'_2=0, \end{aligned}$$

(A20)

$$\begin{aligned}&\mathbf{\tau '}_2 \cdot {\hat{n}}'_1=-\gamma _2 {\hat{n}}'_2 \nabla ' \cdot {\hat{n}}'_2, \end{aligned}$$

(A21)

$$\begin{aligned}&\theta =\theta _B. \end{aligned}$$

(A22)

The stress balance can be resolved into components in the same way as above, using the unit tangent and normal vectors at this interface, ${\hat{t}}'_2$ and ${\hat{n}}'_2$.

At the aqueous–lipid interface $y'=h'_1$, we have surfactant transport and the linear equation of state for the surface tension, respectively:

$$\begin{aligned}&\Gamma '_{t'}+\nabla '_{s_1}\cdot (\Gamma ' u'_1) = D_s \nabla ^{'2}_{s_1}\Gamma ', \end{aligned}$$

(A23)

$$\begin{aligned}&\gamma '_{s1}=\gamma _1 - RT_0\Gamma '. \end{aligned}$$

(A24)

1.3 Osmosis and evaporation

The effect of osmosis is determined by the difference in concentration on either side of the aqueous–corneal interface $y'=0$,

$$\begin{aligned} \mathbf{J'}_0 \cdot {\hat{n}}'_0&= P_c\,(c'-c_0). \end{aligned}$$

(A25)

Here we assume an ideal osmolarity ($c_0=300$ mOsM) on the corneal side of the interface.

To account for the effect of evaporation at the aqueous–lipid interface $y'=h'_1$, we modify a boundary layer model derived thoroughly in Stapf et al. [134]

$$\begin{aligned} J'_e&=\frac{E_0}{1+\frac{k_m}{Dk}h'_2}. \end{aligned}$$

(A26)

Here, Dk represents resistance to evaporation provided by the LL based on its thickness and permeability. Its value comes from a nonlinear least squares fit to clinical data [33] performed by Stapf et al. [134], which is very similar to that of Cerretani et al. [122]. We further modify Dk to include a dependence on the director angle of liquid crystals in the LL. We suggest the orientation of the molecules affects evaporation through

$$\begin{aligned} J'_e&=\frac{E_0}{1+\frac{k_m}{Dk}(0.1+0.9{\cos \theta _B})h'_2}. \end{aligned}$$

(A27)

The maximum evaporation rate occurs when $\theta _B=\pi /2$, which is ten times that of the rate for $\theta _B=0$. The minimum rate matches that used in [134].

1.4 Deriving the leading order system

We nondimensionalize using the scalings in (1), along with

$$\begin{aligned} W'=\frac{K}{\delta ^2 L^2}W, \hspace{25pt}\alpha _i'=\mu _2\alpha _i,\hspace{25pt} \gamma '_t=\mu _2\gamma _t,\hspace{25pt}\gamma '_r=\mu _2\gamma _r. \end{aligned}$$

Then, we asymptotically expand all the dependent variables in powers of the small parameter $\epsilon \ll 1$. For example,

$$\begin{aligned} \Gamma (x,t) \sim \Gamma _0(x,t)+\epsilon \, \Gamma _1(x,t)+\epsilon ^2 \Gamma _2(x,t)+\cdots \end{aligned}$$

We substitute these expansions into the nondimensionalized equations and collect like powers of $\epsilon$. At leading order, we are able to determine the director angle, conservation of mass in both the AL and LL, and conservation of surfactant concentration. At $O(\epsilon )$, we are only able to determine intermediate variables, so we proceed to $O(\epsilon ^2)$. We are then able to determine the AL depth averaged velocity, AL pressure, the force balance equation for the axial velocity in the LL, and conservation of osmolarity. This gives us the closed system of equations from Sect. 2.1. This progression is typical of problems like this [99, 134].